I have made a number of characters that can do everything in a party across a wide variety of role-play systems, and seen several more as both a GM and as a player. I have also seen these characters played to both extremes of what they're capable of, either filling in gaps for other party members and shoring up weak points or turning the co-op game into a one-player game with supporting characters crewed by annoyed players wondering why they're even at the table.
This general type of build ranges from Jack-Of-All-Trades (JOAT) to the Do EveryTHing (DETH). JOATs are fairly common among RPGs, and usually inoffensive (the classic Bard of D&D, the Alchemist of Pathfinder, the shapeshifter of superhero games, etc.), but through powerbuilding the JOAT can evolve into the DETH, which is capable of performing every role required of a party at approximately the same effectiveness as a specialized character.
Now, this in and of itself is not always a problem. Sometimes it is even by design. Mutants & Masterminds includes multiple ways for this effect to occur (the Variable effect and Power Arrays), often resulting in a party of characters who can each handle a wide variety of situations relatively well on their own. Yet, the characters still find cause to join together and collectively tell a story.
Where this becomes a problem is when the DETH character steals the spotlight. They disable the traps instead of the Rogue, charge into battle in front of the Fighter, cast the big blast spells before the Wizard, and pass the speech checks before the Bard player can open their mouth. The obvious answer to this situation is to ask the player to change their character, but you might note that the problem is not with the character. It's how the character is being handled by the player.
In my own experience, I've run into DETH builds that are played as the "fill in" character, such as a powerful Pathfinder alchemist that makes Cure Wounds and Cure Disease potions, assists in combat, has great skills, and uses scrolls and wands to fill in on defensive buffs letting the casters worry about less specific things. This kind of build is obviously a good thing to have around, and I've yet to see a player get upset about someone having their back if they get a bad roll or getting to prep more exciting stuff because they don't need to worry about the basics.
Here are a few examples I've run into, one as a player and the other as a GM:
A fellow player in a currently-running My Hero Academia game in M&M 3e made a character who's quirk was Emotion Motes. Effectively, he gets different powers based on his emotions, with combinations of emotions giving even more powers. Constructs, time stopping, flight, super strength, invisibility, illusions... the list goes on, but I've yet to feel like he's taking over what anyone else does. I do my negative energy, his roleplay adopted brother does his super-strong rage monkey thing, the chain-maker does his Spider-Chain thing, healing mummy girl does her healing mummy thing, and earthquake girl does her earthquake thing.
Because of the nature of the setting - specialized powers taken to extremes - having someone with a lot of options is a really good thing for the party. None of the rest of us do illusions or make the party invisible, or help the party fly, and he's using his abilities to help the rest of the party shine instead of stealing the spotlight. He gets his moments, but this DETH build doesn't leave a bad taste in the other players' mouths.
Another example from M&M 3e in a game I once ran was a character that over-used a power effect called Variable. Variable allows the character to re-allocate Power Points (M&M's currency for character attributes) into new powers, devices, equipment, skills, or anything else reasonable to the character's descriptors on the fly. This is usually how characters as entirely inoffensive as Beast Boy or a mimic are created, but this was a reality warper. The player used the character's reality warping magic to solve absolutely every single problem the character ran into, up to and including teleporting a bunch of invading aliens to another planet (ignoring the character who's primary power was literally "I teleport myself and anything else") and terraforming it for the aliens who specifically asked him not to so they could do it to their liking.
In retrospect I should not have allowed the character to begin with, but the player was a friend of mine and we'd played together several times so I assumed he could be trusted like the first player could. In this case, after the incident with the aliens I asked the player to create a different character. He did, and when I saw it was another flavor of the same idea and asked for a different character again, he got annoyed and left the game. This is the kind of result I'd like to avoid.
In your experience, what is the best way to handle situations of this type? How did it work out?
I am fully aware that all GMs have the power of "ultimate no", but I feel like it gets used too often on DETH builds that would be played as in my first example. I also dislike asking players to play different characters when they spend so much time working on the one, especially since I've been in that situation myself, and if the player is your friend it can ruin your friendship with them. In fact, it did in my second example.
Worse than this, some GMs allow these builds and then try to kill them because they're afraid that they'll be problems. This is antagonistic at best, and can potentially result in an "arms war" of sorts as the under-built characters caught in the crossfire of the over-powered monsters start getting replaced with more optimized characters due to sheer fear of the GM's monsters and the difficulty of the encounters they're facing. This then prompts the GM to make even stronger monsters to combat the stronger PCs.
I also know that answers may be system-specific, and this is perfectly fine. This type of build could occur in most systems, and so I encourage you to answer assuming the mechanics of the system of your choice, though the more generic the concept is, the better (Actions > Action Economy > Standard/Move/Swift Actions > Quickened Spell > Pathfinder Wizard Bonus Feats).